A generalization of pure submodules

Abstract

‎Let $R$ be a commutative ring with identity‎, ‎$S$ a multiplicatively closed subset of $R$‎, ‎and $M$ be an $R$-module‎. ‎The goal of this work is to introduce the notion of $S$-pure submodules of $M$ as a generalization of pure submodules of $M$ and prove a number of results concerning of this class of modules‎. ‎We say that a submodule $N$ of $M$ is emph {$S$-pure} if there exists an $s in S$ such that $s(N cap IM) subseteq IN$ for every ideal $I$ of $R$‎. ‎Also‎, ‎We say that $M$ is emph{fully $S$-pure} if every submodule of $M$ is $S$-pure‎.